Local rigidity of the Euclidean metric for the anisotropic Calder\'on problem

Abstract

We prove local rigidity of the Euclidean metric and close conformally Euclidean ones for the anisotropic Calder\'on problem on smooth compact domains $M\subset\mathbb{R}^n$, $n\ge3$. A smooth Riemannian metric $g$ with the same induced boundary metric and Dirichlet-to-Neumann map as the background $g_0$ is isometric to $g_0$ by a diffeomorphism fixing the boundary pointwise if $\|g-g_0\|_{H^s(M)}$ is sufficiently small, where $s>n/2+1$ is an integer. For $g_0=e^{2c}\e$, we assume that $\|c\|_{C^k(M)}$ is sufficiently small, with an integer $k\ge s+2$ and $k>3n/2+5$.